Modular Arithmetic Calculator

How to use

1. 1Choose an operation: a mod m, modular add, subtract, multiply, a^b mod m, or the modular inverse of a.
2. 2Enter the operand a, the second operand b when the operation needs one, and the modulus m (a positive whole number).
3. 3Read the result, and for plain mod compare the always-non-negative residue with the percent operator remainder and the quotient.
4. 4For the modular inverse, check the line that multiplies a by the inverse to confirm it gives 1 modulo m, or read why no inverse exists.
5. 5Use the GCD panel for any two integers to get the greatest common divisor, the least common multiple, and the Bezout coefficients, then copy any value.

About this tool

Modular Arithmetic Calculator works out the modulo and the standard modular operations that number-theory students, cryptography learners, and programmers keep needing, and it does them exactly for arbitrarily large numbers because every value is held as an arbitrary-precision integer. The starting point is the mod itself. Given a number a and a modulus m, the calculator returns a mod m as the mathematical residue, the value that is always between 0 and m minus 1. This is the detail that trips people up, because the percent operator in C, Java, Go, and JavaScript keeps the sign of the dividend, so in those languages negative-7 percent 3 is negative-1, while negative-7 mod 3 is 2 in proper modular arithmetic. The tool shows both side by side, along with the quotient q in the identity a equals q times m plus the residue, so you can see exactly how the wrap-around works and flags when the two disagree. On top of the plain mod it computes modular addition, subtraction, and multiplication, each reduced back into the 0 to m minus 1 range, and modular exponentiation, a to the power b modulo m, using fast square-and-multiply exponentiation. Square and multiply reduces the running value at every step instead of forming the astronomically large a to the b first, which is why a huge exponent of the kind used in RSA and Diffie-Hellman is handled instantly and never overflows. It also finds the modular multiplicative inverse, the number that multiplies with a to give 1 modulo m, computed with the extended Euclidean algorithm. The inverse exists only when a and m are coprime, meaning their greatest common divisor is 1, and when no inverse exists the tool says so and reports the gcd that blocks it rather than returning a wrong answer, which is also why a prime modulus is so convenient in cryptography. A negative exponent is supported by inverting the base first, where that base is invertible. A separate panel runs the extended Euclidean algorithm on any two integers and reports the greatest common divisor, the least common multiple, whether the pair is coprime, and the Bezout coefficients x and y that satisfy a times x plus b times y equals the gcd, which is the identity the modular inverse falls out of. Inputs accept an optional sign, digit separators, and a 0x or 0b prefix so machine values paste cleanly, and the modulus is required to be a positive whole number. Common uses include discrete mathematics and number theory homework, understanding hashing and checksums, working through RSA and Diffie-Hellman by hand, clock and calendar style wrap-around problems, and debugging the negative-modulo behaviour of a programming language. Everything is computed in your browser with native big-integer math, so the numbers you enter are never uploaded.

Free to use. Works in your browser. No signup, no login.